Environmental Engineering Reference
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where, with
r
PM
as the distance between
P
and
M
,
cos
(
PM
,
n
)
K
(
M
,
P
)=
(7.171)
π
r
PM
and thus,
K
(
P
,
M
)=
−
K
(
M
,
P
)
.
Remark 1
. We demonstrate the transformation from Eq. (7.169) to Eq. (7.170)
by using Dirichlet internal problems as an example. By Theorem 7 in Section 7.8
(Eq. (7.151)), for two-dimensional cases we have
u
(
M
)=
u
(
M
)
−
πτ
(
M
)
,
where the plane double-layer potential
ln
d
s
C
τ
(
)
∂
∂
1
r
PM
u
(
M
)=
P
(7.172)
n
is a harmonic function in the domain both inside and outside of
C
.The
u
(
M
)
stands
for the limit of
u
(
M
)
as
M
tends to
C
from inside of
C
. By Eq. (7.169), we have
u
(
M
)=
u
(
M
)
|
C
=
f
(
M
)
.
Thus
ln
d
s
C
τ
(
∂
∂
1
r
PM
f
(
M
)=
P
)
−
πτ
(
M
)
,
(7.173)
n
where
r
PM
is the normal of vector
PM
or the distance between
P
and
M
.Also,
∂
∂
ln
1
r
PM
∂
ln
r
PM
)
∂
r
PM
∂
=
r
PM
(
−
n
∂
n
∂
1
r
PM
r
PM
∂
)+
∂
r
PM
∂
=
−
cos
(
n
,
x
cos
(
n
,
y
)
x
y
1
r
PM
[
=
cos
(
MP
,
x
)
cos
(
n
,
x
)+
cos
(
MP
,
y
)
cos
(
n
,
y
)]
(
,
)
1
r
PM
(
cos
PM
n
=
−
MP
)
1
·
n
1
=
r
PM
where
(
MP
)
1
and
(
n
)
1
are the unit vectors of
MP
and
n
, respectively. Therefore,
Eq. (7.172) becomes
cos
(
PM
,
n
)
f
(
M
)=
τ
(
P
)
d
s
−
πτ
(
M
)
π
r
PM
C
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